Normal Ordering for Deformed Boson Operators and Operator-valued Deformed Stirling Numbers

The normal ordering formulae for powers of the boson number operator $\hat{n}$ are extended to deformed bosons. It is found that for the `M-type' deformed bosons, which satisfy $a a^{\dagger} - q a^{\dagger} a = 1$, the extension involves a set of deformed Stirling numbers which replace the Stirling numbers occurring in the conventional case. On the other hand, the deformed Stirling numbers which have to be introduced in the case of the `P-type' deformed bosons, which satisfy $a a^{\dagger} - q a^{\dagger} a = q^{-\hat{n}}$, are found to depend on the operator $\hat{n}$. This distinction between the two types of deformed bosons is in harmony with earlier observations made in the context of a study of the extended Campbell-Baker-Hausdorff formula.Comment: 14 pages, Latex fil

Golden gaskets: variations on the Sierpi\'nski sieve

We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, and with a common contraction factor \la\in(0,1). As is well known, for \la=1/2 the invariant set, \S_\la, is a fractal called the Sierpi\'nski sieve, and for \la<1/2 it is also a fractal. Our goal is to study \S_\la for this IFS for 1/2<\la<2/3, i.e., when there are "overlaps" in \S_\la as well as "holes". In this introductory paper we show that despite the overlaps (i.e., the Open Set Condition breaking down completely), the attractor can still be a totally self-similar fractal, although this happens only for a very special family of algebraic \la's (so-called "multinacci numbers"). We evaluate \dim_H(\S_\la) for these special values by showing that \S_\la is essentially the attractor for an infinite IFS which does satisfy the Open Set Condition. We also show that the set of points in the attractor with a unique ``address'' is self-similar, and compute its dimension. For ``non-multinacci'' values of \la we show that if \la is close to 2/3, then \S_\la has a nonempty interior and that if \la<1/\sqrt{3} then \S_\la$ has zero Lebesgue measure. Finally we discuss higher-dimensional analogues of the model in question.Comment: 27 pages, 10 figure

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

Operator renewal theory and mixing rates for dynamical systems with infinite measure

We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates $L^n$ of the transfer operator. This was previously an intractable problem. Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points. In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of $\sum_{j=1}^nL^j$) for the class of systems under consideration. In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for $L^n$ and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws.Comment: Preprint, August 2010. Revised August 2011. After publication, a minor error was pointed out by Kautzsch et al, arXiv:1404.5857. The updated version includes minor corrections in Sections 10 and 11, and corresponding modifications of certain statements in Section 1. All main results are unaffected. In particular, Sections 2-9 are unchanged from the published versio

Team-based learning (TBL): a community of practice

Background Rapid changes in medical practice have a large impact on the demands faced by educators in preparing students for future participation in a multifaceted healthcare workforce. Competencies required by today’s medical graduates encompass the ability to effectively collaborate, communicate and problem solve. The learning needs of medical students have also changed over time. Today’s medical students are highly interconnected, enjoying teamwork and collaborative practice, and desire continuous, explicit feedback. They want structured learning activities, with clear expectations, and enjoy a sense of accomplishment on their achievements. The conflation of these issues has seen many medical schools adopt the model of Team-based learning (TBL). Using the conceptual framework of communities of practice, we sought to qualitatively explore students’ and teachers’ experience of TBL in Year 1 of a graduate entry medical program. Methods Convenience sampling was used to select 169/350 (48%) Year 1 students who completed three TBL sessions. Each TBL session was facilitated by three senior clinicians. Following participation in the TBLs, students were invited to attend focus groups, and all facilitators (n = 9) were invited to attend interviews. A coding framework was developed to code the entire dataset, using the theoretical lens of communities of practice. Results 34/169 (20%) of students attended focus groups. Three facilitators (3/9, 33%) were interviewed. Students and facilitators felt the structure and organisation of TBL made students accountable for their learning and team contributions. The combined expertise and clinical experience of facilitators, with immediate feedback helped groups to work both independently and collaboratively. Facilitators found working with their peers in the TBLs to be a rewarding experience. Conclusions The community of practice found in the TBL classes, provided an enriching and rewarding learning environment that motivated students to build on their basic knowledge and apply what had been learnt. The interactions of experienced, senior clinicians as facilitators, sharing their expertise within a clinical context, prompted effective student engagement in learning and understanding. Our change in curriculum design and pedagogy will assist in preparing medical students for demands of the increasingly complex healthcare systems in which they will work

R-parity Conserving Supersymmetry, Neutrino Mass and Neutrinoless Double Beta Decay

We consider contributions of R-parity conserving softly broken supersymmetry (SUSY) to neutrinoless double beta (\znbb) decay via the (B-L)-violating sneutrino mass term. The latter is a generic ingredient of any weak-scale SUSY model with a Majorana neutrino mass. The new R-parity conserving SUSY contributions to \znbb are realized at the level of box diagrams. We derive the effective Lagrangian describing the SUSY-box mechanism of \znbb-decay and the corresponding nuclear matrix elements. The 1-loop sneutrino contribution to the Majorana neutrino mass is also derived. Given the data on the \znbb-decay half-life of $^{76}$Ge and the neutrino mass we obtain constraints on the (B-L)-violating sneutrino mass. These constraints leave room for accelerator searches for certain manifestations of the 2nd and 3rd generation (B-L)-violating sneutrino mass term, but are most probably too tight for first generation (B-L)-violating sneutrino masses to be searched for directly.Comment: LATEX, 29 pages + 4 (uuencoded) figures appende

Mild sonochemical exfoliation of bromine-intercalated graphite: a new route towards graphene

A method to produce suspensions of graphene sheets by combining solution-based bromine intercalation and mild sonochemical exfoliation is presented. Ultrasonic treatment of graphite in water leads to the formation of suspensions of graphite flakes. The delamination is dramatically improved by intercalation of bromine into the graphite before sonication. The bromine intercalation was verified by Raman spectroscopy as well as by x-ray photoelectron spectroscopy (XPS), and density functional theory (DFT) calculations show an almost ten times lower interlayer binding energy after introducing Br2 into the graphite. Analysis of the suspended material by transmission and scanning electron microscopy (TEM and SEM) revealed a significant content of few-layer graphene with sizes up to 30 $mu$m, corresponding to the grain size of the starting material.Comment: 10 pages 4 figure

Little-Parks effect and multiquanta vortices in a hybrid superconductor--ferromagnet system

Within the phenomenological Ginzburg-Landau theory we investigate the phase diagram of a thin superconducting film with ferromagnetic nanoparticles. We study the oscillatory dependence of the critical temperature on an external magnetic field similar to the Little-Parks effect and formation of multiquantum vortex structures. The structure of a superconducting state is studied both analytically and numerically.Comment: 7 pages, 1 figure. Submitted to J. Phys.: Condens. Mat